Theorem lecm

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	lecm	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\subseteq B\right)\to A{𝐶}_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		sseq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\right)$$
2		breq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)$$
3	1 2	imbi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(A\subseteq B\to A{𝐶}_{\mathscr{H}}B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)\right)$$
4		sseq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
5		breq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
6	4 5	imbi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)\right)$$
7		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
8	7	elimel	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
9	7	elimel	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
10	8 9	lecmi	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
11	3 6 10	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\to A{𝐶}_{\mathscr{H}}B\right)$$
12	11	3impia	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\subseteq B\right)\to A{𝐶}_{\mathscr{H}}B$$